1. Field of the Invention
The invention relates generally to the field friction control. More particularly, the invention relates to control of friction at the micro and nano scale.
2. Discussion of the Related Art
Despite great progress made during the past half century, many problems in fundamental tribology (such as the origin of friction and failure of lubrication) have remained unsolved. Moreover, the current reliable knowledge related to friction and lubrication is mainly applicable to macroscopic systems and machinery and, most likely, will be only of limited use for micro-and nano-systems. Indeed, when the thickness of the lubrication film is comparable to the molecular or atomic size, the behavior of the (film) lubricant becomes significantly different from the behavior of macroscopic (bulk) lubricant [1]. Better understanding of the intimate mechanisms of friction, lubrication, and other interfacial phenomena at the atomic and molecular scales is needed to provide designers and engineers the required tools and capabilities to monitor and control friction, reduce unnecessary wear, and predict mechanical faults and failure of lubrication in micro-electro-mechanical systems (MEMS) and nano-devices [2].
The ability to control and manipulate friction during sliding is extremely important for a large variety of technological applications. The outstanding difficulties in realizing efficient friction control are related to the complexity of the task, namely dealing with systems with many degrees of freedom under strict size confinement, and only very limited control access. Moreover, a nonlinear system driven far from equilibrium may exhibit a variety of complex spatial and temporal behaviors, each resulting in different patterns of motion and corresponding to different friction coefficient [3].
Friction can be manipulated by applying small perturbations to accessible elements and parameters of a sliding system [4-10]. Usually, these control methods are based on non-feedback controls. Recently, the groups of J. Israelachvili [4] (experimental) and U. Landman [5] (full-scale molecular dynamics computer simulation) showed that friction in thin-film boundary lubricated junctions can be reduced by coupling small amplitude (of the order of 1 Å) directional mechanical oscillations of the confining boundaries to the molecular degree of freedom of the sheared interfacial lubricating fluid. Using a surface force apparatus, modified for measuring friction forces while simultaneously inducing normal (out-of-plane) vibrations between two boundary-lubricated sliding surfaces, load- and frequency-dependent transitions between a number of “dynamical friction” states have been observed [4]. In particular, regimes of vanishingly small friction at interfacial oscillations were found. Extensive grand-canonical molecular dynamics simulations [5] revealed the nature of the dynamical states of confined sheared molecular films, their structural mechanisms, and the molecular scale mechanisms underlying transitions between them. Methods to control friction in systems under shear that begin to enable the elimination of chaotic stick-slip motion were proposed by Rozman et al [6]. Significant changes in frictional responses were observed in the two-plate model [7] by modulating the normal response to lateral motion [8]. In addition, the surface roughness and the thermal noise are expected to play a significant role in deciding control strategies at the micro and the nano-scale [9, 10].
Since feedback control methods require specific knowledge of the strength and timing of the perturbations, their application to nano-friction has been very limited. On the other hand, feedback control methods (e.g., proportional feedback) have been applied extensively in many engineering fields. All these feedback controls have been Lipschitzian. Recently, non-Lipschitzian (terminal attractor based) feedback control has been successfully implemented in first order systems such as neural networks [11, 12].
Despite their relative simplicity, phenomenological models of friction at the atomic level [10, 13-16] show a fair agreement with many experimental results using either friction force equipment [7, 18, 19] or quartz microbalance experiments [9, 17, 20]. The basic equations for the driven dynamics of a one dimensional particle array of N identical particles moving on a surface are given by a set of coupled nonlinear equations of the form [16]:m{umlaut over (x)}j+γ{dot over (x)}j=−∂U/∂xj−∂V/∂xj+ƒj+η(t), j=1, . . . N   (1)where xj is the coordinate of the jth particle, m is its mass, γ is the linear friction coefficient representing the single particle energy exchange with the substrate, ƒj is the applied external force, and η(t) is Gaussian noise. The particles in the array are subjected to a periodic potential, U(xj+a)=U(xj), and interact with each other via a pair-wise potential V(xj−xi), j, i=1, 2, . . . N. A system represented by Equation (1) provides a general framework of modeling friction although the amount of detail and complexity varies in different studies from simplified one dimensional models [15, 16, 21, 22] through two dimensional and three dimensional models [17, 23, 24, 25] to a full set of molecular dynamics simulations [25, 26].
Phenomenological models of friction at the atomic level can include the following simplifications (assumptions): (i) the substrate potential is a simple periodic form, (ii) there is a zero misfit length between the array and the substrate, (iii) the same force ƒ is applied to each particle, and (iv) the interparticle coupling is linear. The coupling with the substrate is, however, strongly nonlinear. For this case, using the dimensionless phase variables φj=2πxj/a, the equation of motion reduces to the dynamic Frenkel-Kontorova model [16]{umlaut over (φ)}j+γ{dot over (φ)}j+sin(φj)=ƒ+κ(φj+1−2φj+φj−1).   (2)Without control, Equation (2) exhibits four different regimes: (i) rest (no motion), (ii) periodic sliding, (iii) periodic stick-slip, and (iv) chaotic stick-slip. Different motion types are obtained by only changing the initial conditions of the particle's positions and velocities, but not the system's parameters. The average velocity of the center of mass for the “natural” (i.e., uncontrolled) motion, may take only a limited range of values, namely: (i) v=0 for rest (no sliding), (ii) v=ƒ/γ for periodic sliding motion, and (iii) v=nv0, where n is an integer,
            v      0        =                            2          ⁢          π                          nN          ⁢                                          ⁢          γ                    ⁢                                                  π              -                                                cos                                      -                    1                                                  ⁢                f                                      π                    ⁢                                    (                              κ                -                                  κ                  c                                            )                                      1              /              2                                            ,for periodic stick-slip motion, [16].